College Algebra, Chapter 4, 4.5, Section 4.5, Problem 20

Factor the polynomial $P(x) = x^2 - 8x + 17$, and find all its zeros. State the multiplicity of each zero.

To find the zeros of $P$, we set $x^2 - 8x + 17 = 0$, by using quadratic formula


$
\begin{equation}
\begin{aligned}

x =& \frac{-(-8) \pm \sqrt{(-8)^2 -4(1)(17)}}{2(1)}
\\
\\
=& \frac{8 \pm \sqrt{-4}}{2}
\\
\\
=& \frac{8 \pm 2 \sqrt{-1}}{2}
\\
\\
=& 4 \pm \sqrt{-1}
\\
\\
=& 4 \pm i

\end{aligned}
\end{equation}
$


By factorization,


$
\begin{equation}
\begin{aligned}

P(x) =& x^2 - 8x + 17
\\
\\
=& \left[ x - (4 + i) \right] \left[ x - (4 - i) \right]

\end{aligned}
\end{equation}
$


Therefore, the multiplicity of each zero is $1$.